A necessary and sufficient set of conditions is obtained that relates any two context-free grammarsG1 andG2 with the property that wheneverG2 left—or right—coversG1, the syntax-directed translations (SDT’s) with underlying grammarG1 is a subset of those with underlying grammarG2. Also the case thatG2 left—or right—coversG1 but the SDT’s with underlying grammarG1 is not a subset of the SDT’s with underlying grammarG2 is considered; in this case an algorithm is described to obtain the syntax-directed translation schema (SDTS) with underlying grammarG2 to the given SDTS with underlying grammarG1, if it exists.

The convergence of a proposed second order finite difference method for the determination of an approximate solution of the fourth order differential equationy(4)+fy=g is proved. The matrix associated with the system of linear equations that arises is not even assumed to be monotone, as is often the case in practice. The only requirement is that the functionf(x) be nonnegative. In a typical numerical illustration, the observed maximum errors in absolute value are compared with the respective theoretical error bound for a series of the values of the step size.

A simple algebraic proof of a theorem due to Wigner on the product of three positive matrices is given. It is shown that the theorem holds for four matrices under an additional condition. The proofs are valid in the more general case of operators in a Hilbert space.

A study is made on the pulsatile flow superposed on a steady laminar flow of a viscous fluid in a parallel plate channel rotating with an angular velocity Ω about an axis perpendicular to the plates. An exact solution of the governing equations of motion is obtained. The solution in dimensionless form contain two parametersK2=ΩL2/v which is reciprocal of Ekmann Number and frequency parameter σ=αL2/v. The effects of these parameters on the principal flow characters such as mean sectional velocity and shear stresses at the plates have been examined. For large σ andK2 the flow near the plates has a multiple boundary layer character.

The effect of frictional heat on the temperature distribution in a laminar circular jet has been studied. It is found from the analysis and the graphs that as the Prandtl number decreases from unity the overall temperature difference near the axis of the jet increases but as we move away from the axis it goes on decreasing. The reverse phenomenon happens in the case of increasing Prandtl number.

The character of the equilibrium of a non-viscous, compressible finitely conducting rotating fluid in the presence of a vertical magnetic field along the direction of gravitational field has been investigated. It is shown that the solution is characterised by a variational principle. Based on the existence of variational principle, an approximate solution has been derived for the case of a fluid having exponentially varying density in the vertical direction. Due to finite resistivity of the medium it is found that potentially stable or unstable configuration retains its character. Further the growth rate of disturbance has been obtained corresponding to short and long wavelengths and it is found that electrical resistivity suppresses the growth rate for large wavelengths but it increases the same for small wavelengths. It is further shown that magnetic field has a destabilizing influence for large wavelengths and a stabilizing influence for small wavelengths.

The problem of expressing a general dynamical variable in quantum mechanics as a function of a primitive set of operators is studied from several points of view. In the context of the Heisenberg commutation relation, the Weyl representation for operators and a new Fourier-Mellin representation are related to the Heisenberg group and the groupSL(2,R) respectively. The description of unitary transformations via generating functions is analysed in detail. The relation between functions and ordered functions of noncommuting operators is discussed, and results closely paralleling classical results are obtained.